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\begin{titlepage}
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        % Title
        {\LARGE Companion Book for Stochastic Calculus for Finance}
        
        \bigskip
        % Author(s)
        %\commentout{============================================================
        {Xianghua Gan}%

        {\small Southwestern University of Finance and Economics}
        %=======================================================================}
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\tableofcontents
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\chapter{The Binomial No-Arbitrage Pricing Model}%
\label{cha:the_binomial_no_arbitrage_pricing_model}

\section{One-Period Binomial Model (单时段二叉树模型)}%
\label{sec:one_period_binomial_model}

\textbf{Objective:} Study the pricing of a derivative security.

\noindent \textbf{Model settings:}
\begin{itemize}
    \item If we are going to study the pricing of a \emph{derivative security},
        where do we start from?
        \begin{itemize}
            \item First, we need to define it.
                \item What is it in nature? In nature, it is a random variable.
                \item How to define the random variable? 
                \item This random variable is a function of a security,
                    say, a stock. (Sample space, event, and probability)
        \end{itemize}
    \item Basic elements of the model
        \begin{itemize}
            \item What does the investor's portfolio includes?
                Money, stock, and derivative security.
                \begin{itemize}
                    \item Money: deterministic
                    \item Stock: random
                    \item Derivative: random but usually more risky than stock
                \end{itemize}
            \item How to model \emph{time}?
                \begin{itemize}
                    \item We consider discrete time periods first.
                    \item We consider one period first.
                    \item We index the beginning of the single period as
                        0, and the end of the period as 1.
                \end{itemize}
            \item What are the quantitative variables related to
                money, stock, and derivative?
                \begin{itemize}
                    \item Money: interest rate $r$
                    \item Stock: stock price $S_0$, $S_1$ (What's the
                        difference between $S_0$ and $S_1$?)
                    \item ECO: strike price $K$ at the end
                    \item Should we make some assumptions about these
                        (random) variables?
                \end{itemize}
            \item Can we consider this problem as a decision problem?
                \begin{itemize}
                    \item Objective
                    \item Decisions
                    \item Constraints
                \end{itemize}
        \end{itemize}
\end{itemize}

\noindent \textbf{Solve the model:}
\begin{itemize}
    \item Where do we start?
        \begin{itemize}
            \item Come up some observations, intuition, and thoughts.
            \item Find various ways to tackle this problem.
                \begin{itemize}
                    \item[i)] Suppose the initial wealth is $X_0$
                        at the beginning of the timeline.
                        Then come up a portfolio to replicate the ECO.
                    \item[ii)] Represent the ECO with a portfolio of stock
                        and money.
                        Then compute the initial wealth $X_0$ required.
                \end{itemize}
            % \item Start from the most promising one.
        \end{itemize}
\end{itemize}

\noindent \textbf{Managerial insights}
\begin{itemize}
    \item How to find risk-neutral probabilities such that
        the value of a risky asset keeps the ``same'' as a non-risky asset
        one period afterwards?
    \item In solving this problem, do we use the actual probabilities?
    \item What is the relationship
        between risk-neutral probabilities and actual ones?
\end{itemize}

\section{Multiperiod Binomial Model}%
\label{sec:multiperiod_binomial_model}

\textbf{Model extension: from single period to multiperiod}
\begin{itemize}
    \item What are the same?
        \begin{itemize}
            \item The objective is the same, i.e., replication of the derivative.
            \item The solution approach is the same.
        \end{itemize}
    \item What are the differences?
        \begin{itemize}
            \item From a static problem to a dynamic problem
            \item More decisions required to replicate the derivative
        \end{itemize}
    \item Does the model structure keeps the same in nature?
        \begin{itemize}
            \item Is each binomial leaf independent on other leafs in the same
                time period?
            \item If we can solve each leaf in a given period independently,
                does the problem become simpler?
        \end{itemize}
    \item Do you come up some observations, intuition, and thoughts?
    \item How to prove your conjectures?
\end{itemize}
$S_2^{HH}\equiv S_2(HH)$, which means move $HH$ in $(~)$ to the superscript in the notation.
\begin{align}
    \Delta_1^H S_2^{HH} + (X_1^H - \Delta_1^H S_1^H)(1+r)= & V_2^{HH} \\
    \Delta_1^H S_2^{HT} + (X_1^H - \Delta_1^H S_1^H)(1+r)= & V_2^{HT} \\
    \Delta_1^T S_2^{TH} + (X_1^T - \Delta_1^T S_1^T)(1+r)= & V_2^{TH} \\
    \Delta_1^T S_2^{TT} + (X_1^T - \Delta_1^T S_1^T)(1+r)= & V_2^{TT} \\
    \Delta_0 S_1^{H} + (X_0 - \Delta_0 S_0)(1+r)= & V_1^{TH} \\
    \Delta_0 S_1^{T} + (X_0 - \Delta_0 S_0)(1+r)= & V_1^{TT}
\end{align}

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